 where the subscripts `L' and `M' stand for the laboratory frame and moving frame, respectively; '\(x\) ' denotes the longitudinal (target normal) component; \(\Gamma _d=(1-v_d^2/c^2)^{-1/2}=\cos ^{-1} \theta \) ; \(k_{xL} =k_{0L}\cos \theta \) , where \(\omega _{0}\)  and \(k_0\)  are the laser frequency and wave number, respectively. While the Langmuir frequency remains invariant, the plasma frequency \(\omega _{pe}\)  has a Doppler shift between the two frames [1]}, [2]}, [3]}, i.e.,
\(\omega _{peM}=\Gamma _d^{-1}\omega _{peL}\) 
